Find the limits: (i) $\lim _{x \rightarrow 1}\left[x^{3}-x^{2}+1\right] \quad$ (ii) $\lim _{x \rightarrow 3}[x(x+1)]$ (iii) $\lim _{x \rightarrow-1}\left[1+x+x^{2}+\ldots+x^{10}\right]$.

Step 1: Evaluate the first limit

For a polynomial function, the limit as $x$ approaches a certain value can be found by directly substituting that value into the function. In this case, we substitute $x=1$ into the expression $x^3 - x^2 + 1$.

Step 2: Calculate the first limit

After substituting $x=1$, we perform the arithmetic operations: $1^3 = 1$, $1^2 = 1$. So the expression becomes $1 - 1 + 1$, which simplifies to $1$.

Step 3: Evaluate the second limit

Similarly, for the second limit, we have a polynomial function $x(x+1)$. We can find the limit as $x$ approaches $3$ by directly substituting $x=3$ into the expression.

Step 4: Calculate the second limit

After substituting $x=3$, the expression becomes $3(3+1)$, which simplifies to $3(4)$, resulting in $12$.

Step 5: Evaluate the third limit

The third limit involves a sum of powers of $x$, which is also a polynomial. We substitute $x=-1$ into each term of the sum.

Step 6: Calculate the third limit

When $x=-1$, the terms alternate between $1$ and $-1$. Specifically, $1$, $(-1)^1 = -1$, $(-1)^2 = 1$, and so on. Since there are $11$ terms in total (from $x^0$ to $x^{10}$), and pairs of $(1-1)$ cancel out, the last term $(-1)^{10}$ is $1$, leaving a final sum of $1$.

More from this chapter

• Evaluate the limit as x approaches 0 of (e^(x²) - cos(x)) / (x²).
• Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n² + k²).
• Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).
• Find the derivative of the function given by f(x)=sin (x²).
• Compute the derivative of 6 x^(100)-x^(55)+x.
• lim_x → 0 csc x (√(2 cos² x + 3 cos x) - √(cos² x + sin x + 4)) is:
• Find the derivative of f(x) = x³ - 3x² + 2x - 5 with respect to x.
• Evaluate: (i) lim _x → 1 (x^(15)-1)/(x^(10)-1) (ii) lim _x → 0 (√(1+x)-1)/(x)

All Limits and Derivatives solutions →